In McDuff and Salamon's Introduction to Symplectic Topology, the following open problem is mentioned.
Problem 50 (Standard-at-infinity)
Let $n \ge 3$ and let $\omega$ be a symplectic form on $\mathbb{R}^{2n}$ that agrees with $\omega_0$ on the complement of a compact set. Is $(\mathbb{R}^{2n}, \omega)$ symplectomorphic to $(\mathbb{R}^{2n}, \omega_0)$?
In case it isn't clear, $\omega_0$ is the standard symplectic form. The authors go on to mention some related results:
The same question in dimension four has a positive answer by a celebrated theorem of Gromov [287].
A theorem of Floer–Eliashberg–McDuff [451] asserts that if $(M, \omega)$ is a symplectic manifold with $\pi_2(M ) = 0$ that is symplectomorphic to $(\mathbb{R}^{2n}, \omega_0)$ outside of a compact set, then $M$ is necessarily diffeomorphic to $\mathbb{R}^{2n}$.
Mark McLean and others constructed many examples of symplectic structures on $\mathbb{R}^{2n}$ that are convex at infinity, but are not symplectomorphic to $(\mathbb{R}^{2n}, \omega_0)$ (see McLean [474, 475], Abouzaid–Seidel [6], and Seidel [576]).
Since another open problem concerning the symplectic topology of Euclidean space was recently resolved, I became curious if any progress has been made on the problem stated above. I suppose this will likely be work performed since 2017, since that is when the most recent edition of McDuff–Salamon was published. I am particularly interested in the special case of $\mathbb{R}^6$.
My questions are:
What progress (if any) has been made beyond what is mentioned above?
Is this problem known to be "very hard" by experts in the field?
Does anyone have intuition on whether this is anything close to approachable by an average graduate student?
References
[6] Abouzaid, M., Seidel, P. (2010). Altering symplectic manifold by homologous recombination. http://arxiv.org/abs/1007.3281v3
[287] Gromov, M. (1985). Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae, 82, 307–347.
[451] McDuff, D. (1991). Symplectic manifolds with contact type boundary. Inventiones Mathematicae, 103, 651–671.
[474] McLean, M. (2009). Lefschetz fibrations and symplectic homology. Geometry & Topology 13, 1877–1944. http://arxiv.org/abs/0709.1639
[475] McLean, M. (2012). The growth rate of symplectic homology and affine varieties. Geometric and Functional Analysis, 22, no. 2, 369–442. http://arxiv.org/abs/1109.4466
[576] Seidel, P. (2011). Simple examples of distinct Liouville type symplectic structures. Journal of Topology and Analysis, 3 no. 1, 1–5. http://arxiv.org/abs/1011.0394v2
